First-order Markov model sequence distributions

FIGURE 11.1 B First-order Markov model to represent chain sequence distribution of AMS-AN copolymer. 

The dyad probabilities for copolymer were calculated as a function of reactivity ratios and monomer composition. A first-order Markov model was developed to predict the chain sequence distribution of SAN and AMS-AN copolymers. The six triad concentrations for SAN copolymer were calculated. Nin dyad and 27 triads for random terpolymers were calculated and tabulated in Tables 11.3, 11.5-11.7. 

As mentioned above, a knowledge of the copolymerization parameters enables all the sequence distributions to be calculated. Or inversely, e q)erimentally determined sequence distributions yield the copolymerization parameters. In the simplest case, intensities of appropriate single sequences can be compared. The results are even more reliable, if the determination of the copolymerization parameters is based on the maximal accessible information on the microstructure of the copolymer. To derive the r-parameters from the overall triad distribution is the safest way (13). This may be done by calculating a triad distribution based on a first-order Markov model and optimizing by variation of the reaction probabilities Py imtil the best fit between calculated and experimental data... 

More complex schemes have been proposed, such as second-order Markov chains with four independent parameters (corresponding to a copolymerization with penultimate effect, that is, an effect of the penultimate member of the growing chain), the nonsymmetric Bernoulli or Markov chains, or even non-Maikov models a few of these will be examined in a later section. Verification of these models calls for the knowledge of the distribution of sequences that become longer, the more complex the proposed mechanism. Considering only Bernoulli and Markov processes it may be said that at the dyad level all models fit the experimental data and hence none can be verified at the triad level the Bernoulli process can be verified or rejected, while all Markov processes fit at the tetrad level the validity of a first-order Markov chain can be confirmed, at the pentad level that of a second-order Maikov chain, and so on (10). 

As with the Bernoullian model, comparison between an observed and calculated sequence distribution is required to check for conformity to first-order Markov statistics. Obviously, with only two independent observations, a dyad distribution is insufficient for determining the two independent probabilities of the model. In contrast, a triad distribution provides five independent observations, so this can be used to check conformity to first-order Markov statistics. Trial values of the monomer addition probabilities can be obtained by taking appropriate combinations of the expressions shown in Table 2.3. For example, is given by... 

Several examples of NMR studies of copolymers that exhibit Bernoullian sequence distributions but arise from non-Bernoullian mechanisms have been reported. Komoroski and Schockcor [11], for example, have characterised a range of commercial vinyl chloride (VC)/vinylidene chloride (VDC) copolymers using carbon-13 NMR spectroscopy. Although these polymers were prepared to high conversion, the monomer feed was continuously adjusted to maintain a constant comonomer composition. Full triad sequence distributions were determined for each sample. These were then compared with distributions calculated using Bernoullian and first-order Markov statistics the better match was observed with the former. Independent studies on the variation of copolymer composition with feed composition have indicated that the VDC/VC system exhibits terminal model behaviour, with reactivity ratios = 3.2 and = 0.3 [12]. As the product of these reactivity ratios is close to unity, sequence distributions that are approximately Bernoullian are expected. 

The vast majority of copolymers described in the literature conform to the terminal model for copolymerisation and therefore exhibit sequence distributions which will in principle conform to first-order Markov statistics. Of... 

Within the framework of the above models the problem of the calculation of the sequence distribution is solved in a quite simple way [51-53, 6]. In order to find the probability of any sequence Uk consisting of k units, it should be expressed through the sequence of Markov chain states, the probability of which is calculated usually by means of the routine procedure as a product of the few factors. The first factor 7i corresponds to the initial state Sh and each of the following factors, Vy, corresponds to the transition from the state Sj to Sj at the conditional movement along the sequence of Markov chain states. For instance, in this manner one can calculate the probability of the sequence U3 = S3[1M2M2 in the both cases of terminal model ... 

In this work, coupled SEC-NMR analysis has been demonstrated for three samples of alginates. The NMR data have been treated with two-component 1 order Markov statistical models. The first component reflects a mostly G homopolymer, and the second con onent is a MG copolymer with an almost random or alternating sequence distribution. The relevance of the two components to the epimerization reactions has been noted. 

The microstructure of the SAN copolymer with respect to the chain sequence distribution can be found from the first-order hidden Markov model. The HMM architecture is shown in Figures 11.1A and B. The conditional dyad probabilities... 

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